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arXiv:gr-qc/0510089v1 19 Oct 2005 RECENT RESEARCH DEVELOPMENTS IN PHYSICS Gauge theories of gravity: the nonlinear framework A. Tiemblo and R. Tresguerres Instituto de Matem´ aticas y F ´ isica Fundamental Consejo Superior de Investigaciones Cient ´ ificas Serrano 113 bis, 28006 Madrid, SPAIN (Dated: July 6, 2018) Nonlinear realizations of spacetime groups are presented as a versatile mathematical tool providing a common foundation for quite different formulations of gauge theories of gravity. We apply nonlinear realizations in particular to both the Poincar´ e and the affine group in order to develop Poincar´ e gauge theory (PGT) and metric-affine gravity (MAG) respectively. Regarding PGT, two alternative nonlinear treatments of the Poincar´ e group are developed, one of them being suitable to deal with the Lagrangian and the other one with the Hamiltonian version of the same gauge theory. We argue that our Hamiltonian approach to PGT is closely related to Ashtekar’s approach to gravity. On the other hand, a brief survey on MAG clarifies the role played by the metric–affine metric tensor as a Goldsone field. All gravitational quantities in fact –the metric as much as the coframes and connections– are shown to acquire a simple gauge–theoretical interpretation in the nonlinear framework. PACS numbers: 04.50.+h, 11.15.-q * laetr03@imaff.cfmac.csic.es romualdo@imaff.cfmac.csic.es

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Page 1: arXiv:gr-qc/0510089v1 19 Oct 2005 · 2019-05-06 · arXiv:gr-qc/0510089v1 19 Oct 2005 ... For a gauge group Gto be realized nonlinearly, an auxiliary subgroup H⊂ Gis required to

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RECENT RESEARCH DEVELOPMENTS IN PHYSICS

Gauge theories of gravity: the nonlinear framework

A. Tiemblo∗ and R. Tresguerres†

Instituto de Matematicas y Fisica Fundamental

Consejo Superior de Investigaciones Cientificas

Serrano 113 bis, 28006 Madrid, SPAIN

(Dated: July 6, 2018)

Nonlinear realizations of spacetime groups are presented as a versatile mathematical tool providinga common foundation for quite different formulations of gauge theories of gravity. We apply nonlinearrealizations in particular to both the Poincare and the affine group in order to develop Poincaregauge theory (PGT) and metric-affine gravity (MAG) respectively. Regarding PGT, two alternativenonlinear treatments of the Poincare group are developed, one of them being suitable to deal withthe Lagrangian and the other one with the Hamiltonian version of the same gauge theory. Weargue that our Hamiltonian approach to PGT is closely related to Ashtekar’s approach to gravity.On the other hand, a brief survey on MAG clarifies the role played by the metric–affine metrictensor as a Goldsone field. All gravitational quantities in fact –the metric as much as the coframesand connections– are shown to acquire a simple gauge–theoretical interpretation in the nonlinearframework.

PACS numbers: 04.50.+h, 11.15.-q

[email protected][email protected]

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I. INTRODUCTION

In the search for the unification of forces, different alternatives to Einstein’s original General Relativity have beenproposed [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11], based on the extension of the gauge principle to spacetime groups. Theanalogy existing between gauge theories of gravity and the Yang–Mills theories supporting the standard model allowsto assimilate gravitation to the remaining forces through the characterization of all interactions –gravity included– asmediated by gauge potentials only. In what follows we will focus our attention on Hehl’s Poincare gauge theory (PGT)as much as on metric-affine gravity (MAG), see [12] [13] [14] [15] [16], both formalisms presenting grand adaptabilityin dealing with a diversity of spacetime actions. We claim their value as a suitable support for the unification ofdifferent theoretical points of view on gravitational forces. Indeed, a main result of the present work is to show theclose relationship between the Hamiltonian version of PGT and the Ashtekar approach.

The cornerstone of our treatment consists of nonlinear realizations (NLR’s), a mathematical method [17] [18] [19][20] [21] [22] [23] argued by us to provide a universal foundation for gauge theories of different groups [24] [25] [26][27] [28] [29] [30]. The usefulness of NLR’s in the context of gravitational theories becomes apparent mainly whentranslations are contained in the spacetime gauge group as a subgroup. As a result of the nonlinear approach, thetranslational connections transform into covector-valued 1–forms suitable to be identified as coframes, so that thedynamical gauge theory becomes indistinguishable from spacetime geometry.

For a gauge group G to be realized nonlinearly, an auxiliary subgroup H ⊂ G is required to be chosen in addition.The freedom in selecting the latter provides the nonlinear method with a considerable flexibility. Indeed, a singletheory, say the gauge theory of the Poincare group G, manifests itself in quite different forms depending on thesubgroup H chosen. As we will see, the NLR of PGT with H = Lorentz reveals to be suitable to be taken as thebasis for a Lagrangian approach, whereas the one with H = SO(3) is especially adapted to a Hamiltonian treatment.In the case of MAG, being G the affine group, we also consider two different nonlinear approaches, corresponding tothe choices of the subgroups H = GL(4 , R) and H = Lorentz respectively. The relation between both NLR’s will beexploited to explain the gauge theoretical origin of the ten degrees of freedom of the metric, as much as their Goldsonenature allowing to rearrange them into redefined fields.

The present work is organized as follows. In Section II we briefly review the mathematical foundations of NLR’sin terms of composite fiber bundles. Section III is devoted to what we call the standard approach to PGT, yieldingexplicitly Lorentz covariant coframes and spin connections. The resulting formalism is used to build an Einstein–Cartan Lagrangian description of gravity. Then in Section IV we pay attention to the Hamiltonian approach to PGT.Several technical aspects are discussed, such as a Poincare invariant foliation of spacetime and a general Hamiltonianformalism adapted to exterior calculus. The Hamiltonian dynamics of the Einstein–Cartan action, expressed in termsof real PGT connection variables, is shown to be consistent both with the Lagrangian treatment of Section III andwith an alternative approach of the Ashtekar type which we also develop in some detail. Finally in Section V weapply the nonlinear approach to the affine group to derive metric–affine gravity.

II. FOUNDATIONS OF NONLINEAR GAUGE THEORIES

Principal bundles P (M ,G) describe the structure of ordinary gauge theories of internal Lie groups G. This schemedoes not hold for nonlinear gauge theories, based on the interplay between the gauge group G and a subgroup H ⊂ G.In [31] we invoked composite fiber bundles as the suitable topological background underlying nonlinear realizationsof local symmetries.

Roughly speaking, a composite bundle is a principal bundle P (M ,G) whose G-diffeomorphic fibers are regardedthemselves as bundles whose structure group H is a subgroup of the structure group G of the total bundle. Actually,composite bundles can be built provided a subgroup H ⊂ G exists, whose right action on elements g ∈ G induces acomplete partition of the group manifold G into mutually disjoint orbits gH . By projecting each of these equivalenceclasses to a single element (that is to a left coset) of the quotient space G/H , the group manifold G becomes organizedas a bundle G(G/H ,H) with the orbits gH (diffeomorphic to the subgroup H) as local fibers and with G/H as thebase space. When attached to points of an auxiliary base manifold M , the local bundles G(G/H ,H) constitute thefibers of a composite bundle.

Locally, composite bundles are diffeomorphic to M ×G/H ×H , so that by singling out either the base space M orthe manifold Σ ≃ M ×G/H , two mutually related bundle structures can be recognized in the total bundle space P .On the one hand, the usual bundle structure survives with total G-diffeomorphic fibers projecting to the bundle basespace M . On the other hand, P can also be regarded as consisting of H-diffeomorphic fiber branches attached to themanifold Σ, the latter playing the role of an intermediate base space. The alternative projections π

PM: P → M and

πPΣ : P → Σ become related to each other by defining an additional mapping πΣM

: Σ → M such that the ordinarytotal projection decomposes into two partial projections as π

PM= π

ΣM◦ π

PΣ. Correspondingly, the local sections

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sMP

: M → P decompose as

sMP

= sΣP

◦ sMΣ

. (1)

Let us express the sections introduced in (1) in terms of zero sections –the one associated to sMP

denoted as σMP

and so on–, that is

sMP

= Rg ◦ σMP, g ∈ G , (2)

sΣP= Ra ◦ σΣP

, a ∈ H , (3)

and

sMΣ

= Rb ◦ σMΣ, b ∈ G/H . (4)

As compatibility conditions, they have to satisfy g = b · a and σMP

= Rb−1 ◦ σΣP

◦ Rb ◦ σMΣ(or alternatively

σΣP

◦ Rb = Rb ◦ σΣP). For later convenience we introduce the composite section σξ : M → Σ → P defined from the

total and zero sections in (3) and (4) as

σξ(x) := σΣP

◦ sMΣ

(x) = Rb ◦ σMP(x) . (5)

(The ξ in σξ(x) stands for the parameters labelling the elements b ∈ G/H displayed as Rb in the r.h.s. of (5).) Thereason for introducing (5) is its usefulness for expressing the main results on the nonlinear approach, deduced in [31]and summarized below.

By comparing two bundle elements, both of the form (3), differing from each other by the left action Lg of elementsg ∈ G, in [31] we found the nonlinear transformation law

Lg ◦ σξ(x) = Rh ◦ σξ′(x) , (6)

being Rh the right action of a certain element h ∈ H of the subgroup. For practical reasons, in [30] we transformed(6) into the more manageable formula

g · b = b ′ · h , (7)

where g ∈ G, h ∈ H , and b, b ′ ∈ G/H . Notice that (7) reproduces the original form of the nonlinear law as given in[17]. On the other hand, for infinitesimal g and h = eµ ≈ 1 + µ, a gauge transformation is induced by (6) on fields ψof any representation space of H , namely

δψ(σξ(x)) = ρ(µ)ψ(σξ(x)) , (8)

where ρ(µ) denotes the suitable representation of the H-algebra, see [17] [31].Since we are interested in building the covariant derivatives of the fields ψ transforming nonlinearly as (8), in [30]

we compared the ordinary linear connection, resulting from pulling back the connection 1-form ω by means of σMP

,that is

AM

= σ∗MP

ω , (9)

with the connection characteristic for the nonlinear approach, defined as the pullback of ω by means of σξ, namely

ΓM

= σ∗ξ ω , (10)

the difference between σξ and σMP

being displayed in (5). One finds [30] that the nonlinear connection (10) can beexpressed in terms of the linear one (9) as

ΓM

= b−1( d+AM

) b . (11)

Its gauge transformations, induced by the nonlinear group action (6), are found to be

δΓM

= −( dµ+ [ΓM, µ] ) , (12)

with µ the same H-algebra-valued parameters as in (8). Being ΓM

valued on the Lie algebra of the whole groupG, from (12) one reads out that only those of its components defined on the H algebra behave as true connections

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transforming inhomogeneously, while its components with values on the remaining algebra elements of G/H transformas H-tensors. As a result of (8) and (12), covariant differentials defined as

Dψ := (d+ ρ(ΓM

))ψ , (13)

transform as

δDψ = ρ(µ)Dψ . (14)

Finally, let us make use of the covariant differential operator

D := d+ ΓM, (15)

as read out from (13) without specifying any particular representation, to obtain the field strength as

F := D ∧D = dΓM

+ ΓM

∧ ΓM. (16)

In view of (12) we find (16) to transform as

δF = [µ , F ] . (17)

The relevance of the nonlinear approach for the foundation of gauge theories of gravity becomes evident in thefollowing sections, where we apply it to the local treatment of different spacetime groups.

III. STANDARD NONLINEAR POINCARE GAUGE THEORY

A. Coframes and Lorentz connections

As a first application of the general formalism established in previous section, let us take the group G to be thePoincare group in order to show how its nonlinear local approach gives rise to the Poincare gauge theory of gravity(PGT). Diverse nonlinear realizations are possible depending on the choice of the auxiliary subgroup H ⊂ G. In thissection we take H to be the Lorentz group, yielding an explicitly Lorentz covariant four–dimensional formalism whichprovides the geometrical basis for a Lagrangian approach (developed by us in the language of exterior calculus). Laterin Section IV we present the version of PGT resulting from taking H to be SO(3), suitable to deal with the 3 + 1decomposition underlying PGT Hamiltonian dynamics (to be treated also in exterior calculus, with differential formsplaying the role of dynamical variables).

Our starting point is the fundamental transformation law (6) of nonlinear realizations. After rewriting it in thesimplified form (7), for G = Poincare and H = Lorentz we parametrize the infinitesimal Poincare group element g ∈ Gand the infinitesimal Lorentz group element h ∈ H respectively as

g = ei ǫαPαei β

αβLαβ ≈ 1 + i(ǫαPα + βαβLαβ

), (18)

and

h = ei µαβLαβ ≈ 1 + i µαβLαβ , (19)

where Lαβ are the Lorentz generators and Pµ the translational ones. As read out from (7), the left action of (18) onelements

b = e−i ξαPα (20)

of the coset space G/H , being (20) identical with elements of the group of translations labelled by the finite parametersξα, induces a right action of (19) on

b ′ = e−i (ξα+δξα)Pα . (21)

Replacing (18)–(21) into (7) and taking into account the commutation relations of the Poincare algebra

[Pα , Pβ ] = 0 ,

[Lαβ , Pµ] = i oµ[αPβ] ,

[Lαβ , Lµν ] = −i(oα[µLν]β − oβ[µLν]α

), (22)

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with oαβ as the the Minkowski metric

oαβ := diag (− + ++) , (23)

a simple computation with the help of the Hausdorff-Campbell formula, see Appendix B, yields the value of µαβ in(19) as much as the variation of the translational coset parameters, namely

µαβ = βαβ , δξα = −ββα ξβ − ǫα , (24)

showing ξα to transform exactly as Minkowskian coordinates.On the other hand, using for the linear connection (9) of the Poincare group the notation

AM

= −i(T )

ΓαPα − iΓαβLαβ , (25)

whose components on the Poincare algebra are the linear translational contribution(T )

Γα and the Lorentz one Γαβ

respectively, we find the nonlinear connection (11) to be

ΓM

= −i ϑαPα − iΓαβLαβ , (26)

with the Lorentz connection Γαβ unmodified with respect to the linear case (25), but with the translational connectiontransformed into

ϑα := D ξα +(T )

Γα . (27)

In view of (12), the components of (26) transform respectively as

δϑα = −ϑβββα , δΓαβ = Dβαβ . (28)

The most relevant result is the first equation in (28). According to it, instead of the linear translational connection(T )

Γα in (25) transforming inhomogeneously as δ(T )

Γα = −(T )

Γβββα +Dǫα, we have at our disposal a nonlinear translational

connection 1-form (27) which is Lorentz covector-valued. The latter will be identified from now on as the Lorentzcoframe or tetrad. This feature of deductively providing tetrads with the right transformation properties constitutesone of the main achievements of nonlinear realizations. (Compare with the hypotheses needed to build (27) in thecontext of the linear approach [32].)

B. Gravitational actions and Lagrangian field equations (in the language of exterior calculus)

Let us show that the coframe ϑα and the Lorentz connection Γαβ introduced above are variables suitable to buildPoincare gauge invariant gravitational actions. (Exterior calculus allows to take differential forms as such –rather thantheir components– as dynamical variables. Actually we obtain the field equations by varying a Lagrangian density4–form with respect to the 1–forms ϑα and Γαβ respectively, see below and [16].) The field strengths of the coframeand of the Lorentz connection are found by applying the general expression (16) to the nonlinear Poincare connection(26), yielding

F = −i TαPα − i RαβLαβ . (29)

In (29), the torsion

Tα := Dϑα := dϑα + Γβα ∧ ϑβ (30)

coincides with the translational field strength, while the Lorentzian field strength is the curvature

Rαβ := dΓα

β + Γγβ ∧ Γα

γ . (31)

Both (30) and (31) are building blocks for gravitational actions. For instance, with the help of (31) besides the elementsof the eta basis defined in Appendix A (built from the coframes (27)), one can express the ordinary Einstein-Cartangravitational Lagrange density 4–form with cosmological term as

LEC = − 1

2l2Rαβ ∧ ηαβ +

Λ

l2η , (32)

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see [16] [33]. More general Lagrangians including contributions quadratic in the irreducible pieces of curvature andtorsion (of the form (I)Rαβ ∧ ∗Rαβ , (I)Tα ∧ ∗Tα) are extensively studied in the literature [34] [33]. For the sake ofsimplicity, here we only consider the action S =

∫LEC built from (32). The field equations derived by varying (32)

with respect to the tetrads ϑα are

1

2ηαβγ ∧Rβγ − Λ ηα = 0 , (33)

and on the other hand, variation on the Lorentz connection Γαβ yields

Dηαβ = 0 . (34)

Since Dηαβ = ηαβγ ∧ T γ, from (34) follows the vanishing of torsion, that is

Tα = 0 , (35)

implying that the Lorentz connection reduces to the (anholonomic) Christoffel connection

Γ{}αβ := e[α⌋dϑβ] −

1

2(eα⌋eβ⌋dϑγ)ϑγ . (36)

By replacing (36) in (33), the latter reduces to the standard Einstein vacuum equations with cosmological constantdefined on a Riemannian space. This can be easily checked by translating (33) to the usual Riemannian language ofGeneral Relativity involving the holonomic metric g ij := oαβ ei

αejβ defined from the tetrads ϑα = dxiei

α with theMinkowski metric (23). The anholonomic Christoffel connection (36) transforms into

Γ{}αβ := − dxi e[α

j(∂iejβ] − Γij

kekβ]

)(37)

when reexpressed in terms of the ordinary holonomic Christoffel symbol Γijk := 1

2 gkl(∂ig lj + ∂jg li − ∂lg ij

), while

the curvature (31) with (37) reduces to

Rαβ =1

2dxi ∧ dxl e[αj ekβ]R ilj

k , (38)

being R iljk := 2

(∂[iΓl]j

k+Γ[imkΓl]j

m)

the ordinary Riemann tensor. By inserting (38) into (33), using the definitions

of the Ricci tensor R ij := R ikjk and of the scalar curvature R := g ijR ij and making use of the holonomic version of

the eta basis of Appendix A, being for instance ηj = ηα eαj = 1

3!

√g ǫjklm dxk ∧ dxl ∧ dxm , the Einstein equations

(33) take their standard form

1

2ηαβγ ∧Rβγ − Λ ηα = − eiα

(R ij −

1

2g ij R + Λ g ij

)ηj = 0 . (39)

The fact that the Einstein-Cartan action (32) reproduces the Einstein equations of General Relativity is a test ofthe validity of the PGT approach. However, the latter is flexible enough to be applied to extended gravitationalactions involving quadratic curvature and torsion terms and giving rise to nonvanishing torsion. In general, the useof Poincare gauge variables introduces both, a different perspective in the interpretation of gravity as mediated byconnections only –translational as much as Lorentzian ones– rather than by the base space metric g ij , and moreoverthe possibility of deriving new results which are meaningless in the purely metrical approach. In order to illustratethe latter point, in the next paragraph we show as an example the coupling of gravitational fields (namely ϑα andΓαβ) to fundamental matter fields.

C. PGT invariant action of Dirac fields

If PGT is to be regarded as a basic theory of gravity, one has to understand its coupling to matter beyondphenomenological matter sources. Accordingly, a PGT invariant Dirac action is to be added to PGT gravitationalLagrangians like (32) or its generalizations. To do so, the first step consists in finding the explicit form of the covariantderivative (13) of Dirac bispinors with the Poincare nonlinear connection (26). As shown in [30], a four–dimensional

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realization of the Poincare generators, Pµ as much as Lαβ , can be built from the gamma matrices in the Diracrepresentation

γ0 =

(I 00 −I

), γa =

(0 σa

−σa 0

), γ5 =

(0 II 0

), (40)

being γ5 := i γ0γ1γ2γ3. Besides the usual spinor generators

ρ(Lαβ) = σαβ :=i

8[ γα , γβ ] , (41)

we introduce the finite matrix representation of translational generators as

ρ(Pµ) = πµ :=m

4γµ( 1 + γ5) , (42)

where the dimensional constant m ∼ [L]−1 (in natural units h = c = 1) guarantees the same dimensionality for theintrinsic linear momentum associated to (42) as for the orbital linear momentum −i∂µ, see [30]. Both (41) and (42)provide a nontrivial finite matrix realization of the Poincare algebra (22) in spite of the fact that πµπν = 0. ThePoincare covariant derivative (13) of Dirac fields thus reads

Dψ = dψ − i (Γαβσαβ + ϑµπµ)ψ , (43)

transforming in accordance with (14) as δDψ = i βαβσαβDψ. The PGT–invariant Dirac Lagrange density 4-formbuilt with the help of (43) –without explicit mass term– reads

LD =i

2(ψ ∗γ ∧Dψ +Dψ ∧ ∗γψ) , (44)

where we use the notation of [14], being γ := ϑµγµ and ∗γ its Hodge dual, and as usual ψ := ψ†γ0 and

Dψ := (Dψ)†γ0 = dψ + i ψ(Γαβσαβ + ϑµπµ) . (45)

The Dirac matter action (44) in the presence of gravity has the peculiarity of including the intrinsic translationalcontributions required by the nonlinear gauge approach to PGT, as seen from the covariant derivatives (43) and (45).However, it is interesting to notice that these contributions manifest themselves as a mass term to be added to anexplicitly Lorentz-invariant (rather than Poincare-invariant) Dirac action. Indeed, let us separate the translationalparts of (43) and (45) as

Dψ =: Dψ − iϑµπµψ , Dψ =: Dψ + i ψ ϑµπµ , (46)

where we denote with tildes the translation-independent pieces with the standard form of Lorentz covariant derivatives.By replacing (46) in (44), we realize that the latter transforms into

LD =i

2(ψ ∗γ ∧ Dψ + Dψ ∧ ∗γψ) + ∗mψψ . (47)

To get (47) we made use of the fact that ϑα ∧ ∗ϑβ = δαβ η, with η = ∗1 as the 4-dimensional volume element, seeAppendix A, so that

∗γ ∧ ϑµπµ = −η γµπµ = ∗m (1 + γ5) , (48)

and

− ϑµπµ ∧ ∗γ = −η πµγµ = ∗m (1 − γ5) . (49)

The particular combination of (48) and (49) in the matter action cancels out the γ5 contribution, only remainingthe background mass term in (47). Thus, the nonlinear PGT approach to the coupling of translations to Dirac fieldspredicts the latter ones to be massive.

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IV. HAMILTONIAN TREATMENT OF PGT

A. Remark on the diversity of equivalent nonlinear approaches

Previous section was devoted to a nonlinear approach to the gauge theory of the Poincare group —namely the onewith auxiliary subgroup H = Lorentz— useful to support Lagrangian dynamics of spacetime. However, we recallthat given G =Poincare, the choice of H ⊂ G is not uniquely predetermined. The outline of Section II showed thatnonlinear realizations of a given group G require to fix, in addition to the total symmetry group G itself, a subgroupH ⊂ G enabling the G–gauge transformations to act on representation fields of H . No breaking of the original G–symmetry is needed for it to be realized through explicitly H–symmetric quantities. We are free to select any amongthe available subgroups H ⊂ G in order to construct diverse versions of the gauge theory of one and the same groupG. (Notice that the usual gauge theories of internal groups based on linear realizations rest on the particular NLRcorresponding to the choice H = G.) Nonlinear gauge approaches to a group G corresponding to different auxiliarysubgroups H1, H2 are equivalent to each other, being possible to relate them by means of gauge–like redefinitionsof the fields, as we will show in Section V. Thus descriptions of the local group G with either H1 or H2 constitutedifferent realizations of the same gauge theory.

In the present section we are going to develop a nonlinear local realization of the Poincare group having as auxiliarysubgroup H = SO(3) instead of the Lorentz group considered previously. The resulting SO(3)–covariant formalismreveals to be useful for the Hamiltonian treatment of the PGT approach to gravity. As before, we characterizedynamical variables by means of differential forms. An exterior calculus formulation of Hamiltonian dynamics, fit togauge theories, is briefly outlined in the following. It is mainly based on a proposal by Wallner [35], suitably adaptedto the present nonlinear approach to PGT with H = SO(3).

B. Poincare invariant foliation of spacetime

Nonlinear realizations of G =Poincare with H = SO(3), as derived immediately from the general formalism onNLR’s of Section II, are displayed in Appendix C. The quantities introduced there will be invoked in what follows inthe order needed for our purposes.

First of all, observe that instead of the four–dimensional representation (27) of the tetrad transforming as a Lorentzcovector as shown in (28), in the nonlinear approach of Appendix C we find the coframe ϑα splitted through definition

(C17) into an SO(3) singlet ϑ0 plus an SO(3) covector ϑa , whose explicit gauge transformations are given by (C22)

and (C23) respectively. The invariance (C22) of the time component ϑ0 suggests to perform an invariant foliation ofspacetime into spatial slices as follows.

From the 1–form basis (C17), we define the dual vector basis eα such that eα⌋ϑβ = δβα . Starting from the relation

[eα , eβ ] =(eα⌋eβ⌋d ϑγ

)eγ holding in the 4–dimensional space, the necessary and sufficient condition for a foliation

into 3–dimensional hypersurfaces normal to e0 to exist, according to the Frobenius’ theorem, is that the spatial

restriction of the former formula yields [ea , eb ] =(ea⌋eb⌋d ϑc

)ec , not involving e

0in the r.h.s., or equivalently

ϑ0 ∧ d ϑ0 = 0 . (50)

Notice that, according to (C22), the foliation condition (50) is Poincare invariant. The general solution of (50) reads

ϑ0 = u0 d τ . (51)

In view of (51), let us define uα := ∂τ⌋ϑα such that ∂τ = uα eα = u0 e0

+ ua ea, so that

e0

=1

u0( ∂τ − ua ea ) , (52)

satisfying e0⌋ϑ0 = 1. In what follows, we take (52) as the invariant timelike vector field defining the foliation directionof spacetime. Accordingly, it becomes possible to perform the decomposition of any p–form α [36] into a longitudinaland a transversal part with respect to (52) as

α = ϑ0 ∧ α⊥ + α , (53)

with respective definitions

α⊥ := e0⌋α , α := e

0⌋(ϑ0 ∧ α

). (54)

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Correspondingly, the foliation of the Hodge dual (53) reads

∗α = (−1)pϑ0 ∧ #α− #α⊥ , (55)

where the asterisc ∗ stands for the Hodge dual in four dimensions while # represents its three–dimensional restriction,see [37] [38] [36]. On the other hand, the exterior differential of any p–form decomposes as

dα = ϑ0 ∧[l e

0α− 1

u0d(u0 α⊥

) ]+ dα , (56)

where we introduced the Lie derivative with respect to e0 defined as le0α := d (e0⌋α ) + (e0⌋dα ) , reducing in

particular, for the transversal part α of (53), to

le0α = (e

0⌋dα ) . (57)

The present tetrad–adapted spacetime foliation allows a considerable simplification of the Hamiltonian formalism,mainly when applied to PGT. The obvious reason is that we do not distinguish the foliation direction (determined bythe time–like vector upon which the foliation of spacetime is performed) from the time component (52) of the vectorbasis eα dual to the PGT coframe (C17). That is, the foliation direction is aligned (or even identified) with the gaugeinvariant time vector naturally derived from the PGT approach.

In Appendix D, the present foliation procedure is applied to several quantities defined in Appendix C in the contextof the nonlinear realization of the Poincare group with H = SO(3). In particular, the decomposition into longitudinaland transversal parts is performed, of the different pieces of torsion and curvature necessary for the Hamiltoniantreatment of PGT to be developed next.

C. Hamiltonian formalism in terms of differential forms

Let us outline a Hamiltonian formalism in the language of exterior calculus [35], with the foliation procedure exposedabove incorporated to it. We consider a general gauge theory, its Lagrangian density 4–form

L = L (A , dA ) (58)

depending on the gauge potential 1–form A and on its exterior derivative dA . We require the Frobenius foliationcondition (50) to hold, so that the time component of the tetrad reduces to (51). (In practice, this is equivalent to

work in the time gauge, in which only one degree of freedom of ϑ0 remains different from zero.) Then, in view of (53)and (56), the gauge potential and its differential decompose into longitudinal and transversal parts as

A = ϑ0A⊥ +A , dA = ϑ0 ∧[le

0A− 1

u0d(u0A⊥

) ]+ dA , (59)

respectively. On the other hand, being the Lagrangian density a 4–form, its transversal part vanishes, decomposingsimply as

L = ϑ0 ∧ L⊥ . (60)

From the Lagrangian normal part L⊥ in (60) we define the momenta

#πA

:=∂L⊥

∂(le0A⊥)

, #πA

:=∂L⊥

∂(le0A )

, (61)

and with their help we define the Hamiltonian 3–form

H := u0[le

0A⊥

#πA

+ le0A ∧ #π

A − L⊥

]. (62)

In view of (51), from (62) we reconstruct the Lagrangian density (60) by multiplying by dτ , getting

L = dτ ∧(u0 le

0A⊥

#πA

+ u0 le0A ∧ #π

A −H). (63)

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Variations of (63), with H taken to be a functional of the gauge potentials and their momenta, yield the field equations

u0 le0A⊥ =

δHδ #π

A⊥, u0 le

0A =

δHδ #π

A, u0 le

0

#πA

= − δHδA⊥

, u0 le0

#πA

= −δHδA

. (64)

(As a technical detail, we follow the convention of putting the variations of the generalized coordinates to the left andthose of their conjugate momenta to the right.) On the other hand, the Lie derivative of an arbitrary p–form definedon the 3–space slices and being a functional of the dynamical variables can be expanded as

le0ω = le

0A⊥

δω

δA⊥+ le

0A ∧ δω

δA+

δω

δ#πA

⊥le

0

#πA

+δω

δ #πA∧ le

0

#πA

. (65)

We define generalized Poisson brackets representing the time evolution of differential forms as the expressions resultingfrom substituting the field equations (64) into (65), that is

u0 le0ω = {ω ,H} :=

δHδ #π

A⊥

δω

δA⊥− δω

δ #πA

δHδA⊥

+δHδ #π

A∧ δω

δA− δω

δ #πA∧ δHδA

. (66)

Eq.(66) is a particular case of the more general definition

{α(x), β

(y)}

:=

z

[ ∂ β(y)

∂ #Πi

(z) ∧ ∂ α

(x)

∂Qi(z) − ∂ α

(x)

∂ #Πi

(z) ∧ ∂ β

(y)

∂Qi(z)]∧ η

(z)

(67)

of Poisson brackets for dynamical variables characterized by differential forms [35], where the arbitrary forms α andβ are functionals of the canonically conjugate variables concisely denoted as Qi , #Πi . From (67) we check that thefundamental Poisson brackets satisfy

{Qi(x ) , Qj(y )

}= 0 ,

{#Πi(x ) ,#Πj(y )

}= 0 , (68)

{Qi(x ) ,#Πj(y )

}= δij δ

3(x− y ) , (69)

as expected. Poisson brackets (66) provide the formal instrument needed to calculate the time evolution of dynamicalvariables from the Hamiltonian 3–form (62). Let us mention a few theorems concerning them, useful for practicalcalculations. From definition (66) follows the antisymmetry condition

{ω ,H} = −{H , ω } . (70)

In view of the chain rule of the Lie derivative, that is le0

(σ ∧ ω ) = le0σ ∧ ω + σ ∧ le

0ω , we deduce the distributive

property

{σ ∧ ω ,H} = { σ ,H} ∧ ω + σ ∧ {ω ,H} . (71)

From the normal part of the identity d ∧ dα ≡ 0, namely le0d α− 1

u0 d(u0 le

0α)≡ 0 , it follows

{ dα ,H} − d {α ,H} = 0 , (72)

generalizable to any form defined on the transversal 3–spaces. With these theorems at hand, we are ready to attackthe Hamiltonian dynamics of a PGT gravitational system.

D. Hamiltonian constraints of PGT

As in Section III, we consider the Einstein–Cartan Lagrangian 4–form (32), to which we are going to apply theHamiltonian formalism outlined previously, completed by taking into account the fact that PGT–gravity constitutesa constrained system [39], [40], [41], [42], [43]. We make use of the nonlinear version of PGT of Appendices C andD. Due to the formal identity of definitions (C17), (C18), (C19) with gauge transformations, PGT invariants can bealternatively expressed in terms of quantities corresponding to different NLR’s, so that the PGT invariant action (32)

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11

can be rewritten in terms of the nonlinear variables Γ0a =: Xa , Γab =: ǫabcAc defined in (C18), (C19). The new

form we get for (32) reads

L = − 1

2l2Rαβ ∧ ηαβ +

Λ

l2η = − 1

l2

[DXa ∧ ηa + ϑ0 ∧

(ϑa ∧Ra − Λ η

) ]. (73)

In (73), the components of the four–dimensional nonlinear Lorentz curvature relate to the corresponding SO(3)quantities as shown in (C31), while the SO(3) eta–basis elements are given in (D12). In order to calculate the momentaas defined in (61), we have to find out the normal part of the Lagrangian (73). Making use of the decompositions(D4)–(D10) one gets

L⊥ = − 1

l2

{[ Le

0Xa − 1

u0D

(u0Xa

)]∧ ηa + ϑa ∧ Ra − Λ η

}. (74)

(In (74) and from now on we simplify the notation by suppressing the hat over the SO(3)–valued coframes and basisvectors.) The only nonvanishing momentum obtained from (74) is

#πX

a :=∂L⊥

∂(le

0Xa

) = − 1

l2ηa , (75)

while the remaining ones #πu0

, #πϑa , #π

A⊥

a , #πA

a and #πX

a equal zero. All of them together with (75) constitute theset of primary constraints.

The total Hamiltonian 3–form of a constrained system is built as follows. Starting from the canonical Hamiltonian(62) —adapted in our case to the variables u0, ϑa, Aa

⊥, Aa, Xa⊥, Xa—, we rewrite it, whenever possible, in terms of

covariant expressions, and then we replace the factors multiplying the primary constraints by Lagrange multipliersβi . So we get

H = u0{ 1

l2(Xa

⊥Dηa + ϑa ∧Ra − Λ η )

−Aa⊥

[D#π

A

a + ηabc(Xb

⊥#π

X⊥

c +Xb ∧ #πX

c + ϑb ∧ #πϑc

) ]}

+βu0#πu0

+ βaϑ ∧ #πϑ

a + βaA

#πA

a + βaA∧ #π

A

a + βaX

#πX

a

+βaX∧(

#πX

a +1

l2ηa

). (76)

From (76), the time evolution of any dynamical variable is calculable in principle with the help of Poisson brackets of

the form (66), suitably generalized to the whole set of conjugate variables u0 ,#πu0

; ϑa ,#πϑa ; Aa

⊥ ,#π

A⊥

a ; Aa ,#πA

a ;

Xa⊥ ,

#πX

a ; Xa ,#πX

a .Primary constraints are required to be stable. That is, their respective evolutions in time are enforced to vanish,

giving rise to four secondary constraints plus two conditions on the Lagrange multipliers as follows. On the one hand,the evolution equations

u0 le0

#πu0

= − 1

l2

(0)

+Xa⊥ ϕ

(3)

a

)+Aa

⊥ ϕ(1)

a , (77)

u0 le0

#πA

a = u0 ϕ(1)

a , (78)

u0 Le0

#πA

a = − 1

l2ϕ

(2)

a , (79)

u0 Le0

#πX

a = −u0

l2ϕ

(3)

a , (80)

when put equal to zero, yield the secondary constraints

ϕ(0)

:= ϑa ∧Ra − Λ η , (81)

ϕ(1)

a := D#πA

a + ǫabc(Xb

⊥#π

X⊥

c +Xb ∧ #πX

c + ϑb ∧ #πϑc

), (82)

ϕ(2)

a := D(u0 ϑa) + u0Xb⊥ϑb ∧ ϑa , (83)

ϕ(3)

a := Dηa , (84)

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while on the other hand the vanishing of the time evolution of the remaining primary constraints, namely

u0 Le0

(#π

X

a +1

l2ηa

)= − 1

l2ηab ∧

(βbϑ + u0Xb

), (85)

u0 Le0

#πϑa = − 1

l2

{[βb

X−D

(u0Xb

) ]∧ ηab + u0

(Ra − Λ ηa

)}, (86)

fixes conditions on the Lagrange multipliers βaϑ and βa

Xrespectively. By solving (85) and (86) equaled to zero we get

βaϑ = −u0Xa , (87)

as much as

βaX

= D(u0Xa

)+ u0

[#Ra − ea⌋

(ϑb ∧ #Rb

)− 1

2ϑa #

(ϑb ∧Rb

)+

Λ

2ϑa

]

≈ D(u0Xa

)+ u0

[#Ra − ea⌋

(ϑb ∧ #Rb

) ]. (88)

The simplification in (88) follows from taking ϕ(0)

in (81) into account. (The symbol ≈ indicates that the equationholds weakly, that is in the subspace of the phase space where all constraints hold.)

Let us deduce several consequences of the secondary constraints (82)–(84). Having in mind also the primary ones,

the constraint (82) reduces weakly to ϕ(1)

a ≈ − 1l2ϑa ∧ ϑb ∧Xb ≈ 0 implying ϑa ∧Xa ≈ 0 . On the other hand, the

vanishing of (84) implies ϕ(3)

a := Dηa = ηab∧Dϑb ≈ 0 . Replacing here Dϑb ≈ −(d log u0 +Xa

⊥ ϑa)∧ϑb as deduced

from the constraint (83), one gets(d log u0 +Xb

⊥ ϑb)∧ ηa ≈ 0 , proving that d log u0 + Xa

⊥ ϑa ≈ 0 is a constraintby itself, so that Dϑa, being proportional to it, also vanishes weakly. In summary, (82)–(84) with the help of theprimary constraints give rise to

d log u0 +Xa⊥ ϑa ≈ 0 , ϑa ∧Xa ≈ 0 , D ϑa ≈ 0 , (89)

whose geometrical meaning as the vanishing of several torsion pieces becomes clear by comparison with (D1), (D2).On the other hand, (88) can be further simplified. In view of the definition of Ra in (D10), we put ϑa ∧ #Ra =

ϑa ∧ #F a + 12

[ea⌋eb⌋

(ϑd ∧Xd

) ]ǫabc

#Xc , where the last term vanishes according to the second equation in (89),

and so does the first term of the r.h.s. since the covariant differential of the third equation in (89) yields DDϑa =

ηab ∧ F b = ϑa ∧ ϑb ∧ #F b ≈ 0 , so that ϑa ∧ #Fa ≈ 0 . Thus we conclude

ϑa ∧ #Ra = 0 . (90)

Replacing (90) in (88), the latter reduces to its ultimate form

βaX≈ D

(u0Xa

)+ u0 #Ra . (91)

According to the general treatment of constrained systems [39] [40] [41], we furthermore have to require the secondaryconstraints (81)–(84) to be stable in time. The time evolution of (82) is found to automatically satisfy

u0 Le0ϕ(1)a ≈ 0 , (92)

by simply taking into account the already known constraints. However, new conditions on the Lagrange multipliersare necessary to guarantee the stability of (83) and (84). For the latter we find

u0 Le0ϕ(3)a ≈ ηab ∧

(Dβb

ϑ − ηbc ∧ βcA

), (93)

implying, when enforced to vanish,

βaA

= #Dβaϑ − ea⌋

(ϑb ∧ #Dβb

ϑ

)− 1

2ϑa #

(ϑb ∧Dβb

ϑ

), (94)

which in view of (87) with (89) reduces to

βaA≈ u0

[ǫabcX

b⊥X

c − #DXa + ea⌋(ϑb ∧ #DXb

) ]. (95)

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On the other hand, time evolution of (83) is calculated to be

u0 Le0ϕ(2)a ≈ −u0 ϑa ∧

[d

(βu0

u0

)+ βb

X⊥

ϑb +Xb⊥ β

ϑb

], (96)

whose vanishing ensures the stability Le0

(d log u0 +Xa

⊥ ϑa)≈ 0 of the first equation in (89). Finally we require the

stability of the constraint (81). Making use of (87) and (91) we find

u0 le0ϕ(0) ≈ − 1

u0d[u0

(ϑa ∧ βa

A− u0Xa

⊥ ηab ∧Xb) ]

. (97)

The differentiated quantity in (97) can be calculated in view of (95) with (89), yielding

ϑa ∧ βaA− u0Xa

⊥ ηab ∧Xb ≈ u0 #DXa + u0Xa⊥

#[ea⌋

(ϑb ∧Xb

) ]≈ u0 #DXa , (98)

so that (97) transforms into

u0 le0ϕ(0) ≈ − 1

u0d[

(u0)2 ϕ(4)], (99)

where we defined

ϕ(4) := ϑa ∧ #DXa . (100)

The vanishing of (99) is the stability condition of (81), so that in principle (99) should be taken as a new constraint.However, one can check that for (99) to be stable, ϕ(4) as defined in (100) must vanish, so that (100) itself, ratherthan the less restrictive condition (99), is to be considered as the new constraint.

In view of the vanishing of (100), the Lagrange multiplier (95) reduces to

βaA≈ u0

(ǫabcX

b⊥X

c − #DXa), (101)

and finally one can prove that the constraint (100) is stable, thus completing our search for the constraints (and forthe solved Lagrange multipliers) of the theory.

E. Comparison to the Lagrangian approach to PGT

The meaning of the gravitational equations obtained in the context of the Hamiltonian approach to PGT becomesclarified by comparing them with the ordinary PGT Lagrangian equations (33), (35) derived from the same action(32). Besides the constraints of previous subsection, we have to consider the evolution in time of the spatial triad ϑa,of the SO(3) connection Aa and of the nonlinear boost connection Xa. As we know, their Lie derivatives along thetime direction e

0are found from (76) with the help of the Poisson brackets (66). Taking into account the previously

calculated values of the Lagrange multipliers (87), (91) and (101), and the definitions of covariantized Lie derivativesin (D3), (D9) and (D5), we get

u0 Le0ϑa = βa

ϑ ≈ −u0Xa ⇒ Le0ϑa +Xa ≈ 0 , (102)

u0 F a⊥ = βa

A≈ u0

(ǫabcX

b⊥X

c − #DXa)⇒ Ra

⊥ ≈ −# (DXa) , (103)

u0 Le0Xa = βa

X≈ D

(u0Xa

)+ u0 #Ra ⇒ Le

0Xa − 1

u0D

(u0Xa

)≈ #Ra . (104)

The evolution equations (102)–(104) plus the set of constraints found above summarize the PGT Hamiltonian dynamicswe want to compare with the Lagrangian field equations (33), (35). In order to do so, we decompose the latter ones intotheir longitudinal and transversal parts making use of the results of Appendix D. Let us begin with the Lagrangianresult (35) of vanishing torsion, which in view of (D1), (D2), takes the form

0 = T 0 = −ϑ0 ∧(d log u0 +Xa

⊥ϑa

)+ ϑa ∧Xa , (105)

0 = T a = ϑ0 ∧(

Le0ϑa +Xa

)+D ϑa , (106)

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where we reintroduced the hat notation of Appendices C and D in order to avoid confusions. It is easy to checkthat the four equations contained in (105), (106) coincide with the Hamiltonian constraints (89) together with theevolution equation (102). The Hamiltonian equations involved acquire in this way an explicit geometrical meaning.On the other hand, the Einstein equations (33) decompose as the time component

0 =1

2η0βγ ∧ Rβγ − Λ η0 = −ϑ0 ∧

(ϑa ∧Ra

)+(ϑa ∧Ra − Λ η

), (107)

and the spatial components

0 =1

2ηaβγ ∧ Rβγ − Λ ηa = −ϑ0 ∧

{[ Le

0Xb − 1

u0D

(u0Xb

) ]∧ ηab + Ra − Λ ηa

}− ηab ∧DXb , (108)

respectively. Regarding (107), notice that the transversal part is the constraint (81), while the vanishing of thelongitudinal part follows from (103) with the constraint (100). Furthermore, (100) also yields the vanishing of the

transversal part of (108) since trivially 0 ≈ ϑa∧ϕ(4)

= ϑa∧ ϑb∧#DXb = ηab∧DXb , where we made use of the Hodgedual relations of Appendix A, suitably adapted to three–dimensional space. The vanishing of the longitudinal part of(108) follows from performing the exterior product of (104) by ηab. The coincidence of the resulting expression with the

one in (108) can be easily proved as follows. We start with the three–dimensional-adapted identity #Rb∧ηa ≡ ϑa∧Rb,see Appendix A, and contract it with eb. Then, by invoking the constraints (81) and (90) together with the identity#(α∧ ϑb) ≡ (eb⌋#α), see Appendix A, we get #Rb ∧ ηab ≈ −(Ra−Ληa ). Thus we were able to deduce the completeset of Lagrangian equations (105)–(108) from the Hamiltonian approach to PGT.

Observe that the reciprocal derivation is not possible. Indeed, the longitudinal part of (107), that is ϑa∧Ra⊥ ≈ 0, is

obtained as the trace of (103) provided (100) vanishes. However, equation (103) itself has not a Lagrangian equivalent.Something similar can be said about (104) and the longitudinal part of (108). It is precisely the presence of (103) and(104) that makes it possible to put the Hamiltonian Einstein equations together into a very simple SO(3)–covariantformulation on four–dimensional spacetime. Taking into account (D4) and (D8), both (104) and the Hodge dual of(103) rearrange into

DXa − ∗Ra ≈ 0 , (109)

while the trace ϑa ∧Ra⊥ ≈ 0 of (103) besides the constraint (81) are summarized by the four–dimensional formula

ϑa ∧Ra − Λ η0 ≈ 0 , (110)

being η0

:= e0⌋η =: η , see (D12). Equations (109), (110) with the additional conditions (105), (106) of vanishing

torsion constitute the condensed form of the Hamiltonian PGT equations derived from the ordinary Einstein–Cartanaction (32) in vacuum.

F. Relation to Ashtekar variables

We are interested in calling attention on the close relationship in which the variables (C17)–(C19) introduced byus in the context of Hamiltonian PGT stand to the Ashtekar variables, see [10] [11] [44] [36] [45]. To make the linkapparent, notice that the Lagrangian dynamics of the Einstein–Cartan action (32) is not modified by adding to it aterm as

L = LEC − β1

2l2R∗

αβ ∧ ηαβ = − 1

2l2

(Rαβ + β R∗

αβ

)∧ ηαβ +

Λ

l2η , (111)

the latter being proportional (with arbitrary constant coefficient β ) to the Lie dual of the curvature, that is to

R∗αβ :=

1

2ηαβ

µν Rµν , (112)

(not to be confused with the Hodge dual considered in Appendix A). Indeed, as compared with the Lagrangianapproach of Subsection III. B, the contributions of the additional term in (111) to the field equation analogous to(34) still imply zero torsion (35), while the modification of the Einstein equation (33) as deduced from (111) is enlarged

by a term proportional to Rµν ∧ ϑν ≡ −DTµ , which also vanishes for vanishing torsion. So the Lagrangian dynamics

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derived from (111) is indistinguishable from the one obtained from (32). We will show that also the Hamiltonianequations coincide with those of the standard Einstein–Cartan case.

Let us start by reexpressing (111) in terms of the PGT variables (C17)–(C19) making use of (C31), so that themain term in (111) becomes

1

2

(Rαβ + β R∗

αβ

)∧ ηαβ =

(ϑ0 ∧ ϑa + β η0a

)∧Ra +

(β ϑ0 ∧ ϑa − η0a

)∧DXa . (113)

Then we combine the PGT connection fields (C18), (C19), namely the SO(3) connections Aa and the nonlinear boostconnections Xa, into a modified SO(3) connection

Aa := Aa + β Xa , (114)

which we claim to be a variable of the Ashtekar type, as will be justified by the following development. For laterconvenience, in (114) the constant β is chosen to be the same as in (111), without further determining its value [46],non even prejudging for the moment about its real or complex character. The SO(3) field strength built from (114)reads

F a := d Aa +1

2ǫabc A

b ∧ Ac = Ra + β DXa + (β2 + 1 )1

2ǫabcX

b ∧Xc , (115)

compare with (C29), (C30). Replacing (115) in (113) (as much as the covariant derivative DXa in (115), defined as

(C29), by DXa in terms of (114)), we get

1

2

(Rαβ + β R∗

αβ

)∧ηαβ =

(ϑ0 ∧ ϑa + β η0a

)∧Fa−(β2+1 )

[η0a ∧D Xa+

(ϑ0 ∧ ϑa − β η0a

)∧ 1

2ǫabcX

b∧Xc]. (116)

We are free to maintain the value of β arbitrary or even to choose it to be real despite the Lorentzian signature weare dieling with [46], but it is obvious that a major simplification of (116) follows from taking β2 = −1 . In particularwe fix β = i , so that (111) becomes an action of the Jacobson–Smolin type [44], namely

L = − 1

l2(− )Rαβ ∧ ηαβ +

Λ

l2η = − 1

l2(ϑ0 ∧ ϑa + i η0a

)∧ Fa +

Λ

l2η , (117)

with the anti–self–dual curvature defined from the curvature and its Lie dual (112) as

(− )Rαβ :=1

2

(Rαβ + i R∗

αβ

), (118)

thus satisfying (− )R∗αβ = −i (− )Rαβ . The components of the corresponding anti–self–dual connection

(− )Γαβ :=1

2

(Γαβ + i Γ∗

αβ

), with Γ∗

αβ :=1

2ηαβ

µν Γµν , (119)

relate to the complex Ashtekar connection (114) with β = i as

Aa := Aa + iXa = ǫabc(− )Γbc = 2i (− )Γ0a , (120)

while the field strength (115) reduces to

F a = Ra + iDXa . (121)

For what follows, it is convenient to rewrite (117) making use of the identity ϑ0∧ϑa∧ Fa ≡ ∗Fa∧ η0a , see AppendixA, together with the notation of (D12), as the complex Lagrangian

L =1

l2

[ (iF a − ∗F a

)∧ ηa + Λ ϑ0 ∧ η

], (122)

to which we proceed to apply the Hamiltonian treatment developed previously. The normal part of (122) reads

L⊥ =1

l2

[ (i F a

⊥ − #Fa)∧ ηa + Λ η

], (123)

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16

where we used definitions

F a⊥ := le

0A

a − 1

u0D

(u0 Aa

), F

a:= d A

a+

1

2ǫabc A

b ∧ Ac, (124)

analogous to those in (D9). All momenta calculated from (123) result to be primary constraints, being the onlynonvanishing one

#πA

a :=∂L⊥

∂(le

0A

a) =

i

l2ηa , (125)

compare with (75), whereas the remaining ones #πu0

, #πϑa , #π

A⊥

a are all equal to zero. Proceeding as in SubsectionIV. D, we build the total Hamiltonian 3–form

H = u0[ 1

l2

(ϑa ∧ F

a − Λ η)− Aa

(D#π

A

a + ǫabc ϑb ∧ #πϑ

c

) ]

+βu0#πu0

+ βaϑ ∧ #πϑ

a + βa

A⊥

#πA

a + βa

A∧(

#πA

a − i

l2ηa

). (126)

Time evolution of any dynamical variable is calculable with the help of the Poisson brackets (66) adapted to the

conjugate variables u0 ,#πu0

; ϑa ,#πϑa ; Aa

⊥ ,#π

A⊥

a ; Aa,#π

A

a . Repeating the steps of Subsection IV. D, the stabilityconditions of the primary constraints yield on the one hand the secondary constraints

ϕ(0)

:= ϑa ∧ Fa − Λ η , (127)

ϕ(1)

a := D#πA

a + ǫabc ϑb ∧ #πϑ

c , (128)

and on the other hand the conditions on the Lagrange multipliers

i ηab ∧ βbϑ − D (u0 ϑa) = 0 , (129)

i ηab ∧ βb

A+ u0

(F a − Λ ηa

)= 0 . (130)

From (129) follows

βaϑ = −i

[ea⌋#D (u0 ϑb)

]ϑb +

i

2ϑa #

[D (u0 ϑb) ∧ ϑb

], (131)

and from (130) with (127)

βa

A≈ i u0

[#F

a − ea⌋(ϑb ∧ #F

b) ]

. (132)

The constraint (128) is stable. Instead, (127) requires the additional stability condition

βaϑ ∧

(F a − Ληa

)+ ϑa ∧ D βa

A≈ 0 , (133)

which, by making use of (129), (130) and (132), transforms into

d[

(u0)2(ϑa ∧ #F

a) ]

≈ 0 . (134)

The stability of (134) requires

ϑa ∧ #Fa ≈ 0 , (135)

constituting a new constraint, replacing the –less restrictive– previously found (134). Substitution of (135) in (132)yields

βa

A≈ i u0 #F

a. (136)

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Our search for constraints finishes by checking that (135) is stable.Let us at this point argue in favor of the equivalence between the Ashtekar and the Hamiltonian PGT approach to

gravity. The proof requires first to support the strict Ashtekar character of the present treatment; we achieve it byshowing that, although somewhat hidden by the exterior calculus notation, the already obtained constraints satisfiedby the complex variables (120) coincide with the Ashtekar constraints. The second step consists in demonstratingthat the complex approach in terms of (120) –that is, in terms of variables of the Ashtekar type built from PGTquantities– constitutes an alternative formulation of the real Hamiltonian approach to PGT as presented in IV. D,

E. Actually, we are going to show that, by decomposing the dynamical equations of the Ashtekar kind into their realand imaginary parts, they reproduce the Hamiltonian PGT equations.

Our first task is to rewrite the constraints (127), (128) and (135) in a language suitable to reveal them as the wellknown Ashtekar constraints. (When comparing the following results with the standard equations, for instance (4) and(6) of [47], the reader must have in mind the interchanged role of the latin letters in Ashtekar’s notation for indicesas compared with ours, being in our case those of the beginning of the alphabet reserved for internal SO(3) indices,see Appendix C, while those of the middle of the alphabet are assigned by us to the general coordinate indices ofthe underlying four–dimensional manifold.) Let us begin with the constraint (128), transforming with the help of the

primary constraints #πϑa and (125) into ϕ

(1)

a ≈ il2D ηa. Its 3–dimensional Hodge dual manifests itself as the Gauss

law

Ga := #(D ηa

)=

1

eDi

(e ea

i)≈ 0 , (137)

with e as the determinant built from the components of the triad ϑa = dxieia. Similarly, from (135) we get

#(ϑa ∧ #F

a)

= ϑbF aba = dxiF a

ia ≈ 0 ⇒ Vi := eaj Fij

a ≈ 0 , (138)

where one recognizes Ashtekar’s vector constraint. Finally, the Hodge dual of (127) becomes the ordinary scalarconstraint, namely

S := #(ϑa ∧ F

a − Λ η)

=1

2ǫa

bc F abc − Λ =

1

2ǫa

bc ebi ec

j F aij − Λ ≈ 0 . (139)

In view of (137)–(139), the full identification of (120) with the Ashtekar variables will be complete once the spinconnection, and thus (120) itself, becomes entirely determined by the coframe as a consequence of the vanishing oftorsion, as will be shown below.

On the other hand, the announced proof of the exact coincidence between the present results and the PGT onesin IV. D, E requires to reproduce here the dynamical equations (109) and (110) together with the zero torsion

conditions (105), (106). We proceed as follows. From (126) we calculate the evolution equations for Aa

to be

u0 le0A

a= βa

A+ D (u0A⊥) . (140)

Taking the value (136) into account with the first definition in (124), from (140) we get

F a⊥ = i#F

a. (141)

Equation (141) can be rewritten in 4–dimensional notation by recalling

F a = ϑ0 ∧ F a⊥ + F

a, ∗F a = ϑ0 ∧ #F

a − #F a⊥ , (142)

according to (53) and (55) respectively. So, in four dimensions, (141) transforms into

∗F a = −i F a , (143)

establishing a simple relation between the field strength and its Hodge dual. (By the way, notice that (143) guarantees

the automatic fulfilment of the gauge theoretical equation D ∗F a = −i DF a ≡ 0.) Furthermore, (127) and (141) with(135) yield

ϑa ∧ F a − Λ η0

= 0 . (144)

Let us show that (143) and (144) constitute an alternative way to display the previously found PGT equations (109)and (110) respectively. Indeed, taking (121) into account one checks that (143) is a shorthand for (DXa − ∗Ra ) −

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18

i ∗ (DXa − ∗Ra ) ≈ 0 , doubly reproducing (109), while (144) can be rewritten as(ϑa ∧Ra − Λ η

0

)−i ϑa∧DXa ≈ 0 .

The imaginary contribution, reexpressed in terms of the torsion components (C27), (C28), reads ϑa∧DXa ≡ −dT 0 +

Ta ∧Xa , so that provided the torsion vanishes, (144) reproduces (110). If this is the case, the dynamical equationsderived from both approaches coincide.

The result of zero torsion follows in fact on the one hand from (128), reduced in view of the primary constraints#πϑ

a ≈ 0 and (125) to ϕ(1)

a ≈ il2D ηa = i

l2ηab ∧ D ϑb ≈ 0, and on the other hand from (129) with the value of βb

ϑ given

by the evolution equation for the triad, namely βbϑ = u0 Le

0ϑb . Combining both results into a single four–dimensional

expression in order to facilitate calculations, we get

1

u0ϑ0 ∧

[i ηab ∧ u0 Le

0ϑb − D (u0 ϑa)

]− i ηab ∧ D ϑb = D

(ϑ0 ∧ ϑa + i η0a

)≈ 0 , (145)

with D as the SO(3) covariant derivative built with the complex connection (120). Taking into account the expressions(C27) and (C28) for torsion, we find

0 ≈ D(ϑ0 ∧ ϑa + i η0a

)= T 0 ∧ ϑa − ϑ0 ∧ T a + i η0ab ∧ T b , (146)

whose unique solution is the vanishing of the whole torsion Tα .Observe that zero torsion allows to simplify (131) enormously provided one expresses it as

βaϑ = −u0Xa − i ea⌋#d u0 , (147)

that is, in terms of the real and imaginary parts of (120) separately, rather than in terms of the whole complexconnection (120). Compare (147) with (87).

A more relevant consequence of Tα ≈ 0 is that (120) becomes expressible in terms of the torsion free connection

Γ{}µν of the form displayed in (36) as

Aa = − i

2

[eµ⌋eν⌋

(ϑ0 ∧ ϑa + i η0a

) ]Γ{}µν =

1

2ǫabcΓ

bc{} + i Γ0a

{} , (148)

see (C18), (C19). This completes the correspondence between Hamiltonian PGT built exclusively in terms of realquantities as developed before, and the present Hamiltonian treatment in terms of complex Ashtekar variables, thelatter satisfying the Ashtekar constraints (137)–(139) and being built from coframes as shown in (148). We claimthat the identification of the Ashtekar complex connection as the combination (120) of the PGT real fields (C18) and(C19) allows to regard both Hamiltonian approaches to gravity –Ashtekar’s and PGT– as alternative reformulationsof each other.

V. METRIC-AFFINE GRAVITY

Finally, let us briefly illustrate the nonlinear techniques when applied to a spacetime group other than the Poincaregroup. We consider in particular the affine group giving rise to metric–affine gravity (MAG) [16] [24] [25] [26] [29],which constitutes an open and active research field –see [16] [48] and references therein– proposed as an alternativeto more usual descriptions of gravity. In the various nonlinear approaches to PGT studied in previous sections, weremarked the interpretation of tetrads as gauge–theoretical quantities, specifically as nonlinear translative connections.This result remaining valid in the context of the MAG theory to be presented here, we are going to pay further attentionto the origin of the degrees of freedom of the MAG–metric, which also turn out to be of gauge–theoretical natureas Goldstone fields. To make this point apparent, we consider two different nonlinear approaches to the affine groupG = A(4 , R), corresponding to the choices of the auxiliary subgroup either as the general linear group H1 = GL(4 , R)or as the homogeneous Lorentz group H2 = SO(3 , 1), and then we relate them to each other. The reason for doingso is that by simply applying the standard nonlinear gauge procedure to the affine group with H1 = GL(4 , R), nometric tensor becomes manifest. For the latter to be deduced, the formalism obtained for H1 = GL(4 , R) has to becompared to the one derived for H2 = SO(3 , 1), as will be shown immediately.

Let us start with the nonlinear realization of G = A(4 , R) with H1 = GL(4 , R). Proceeding as usual, we replacein the simplified form (7) of (6) the suitable G elements

g = ei ǫαPαei ωα

βΛαβ ≈ I + i ǫαPα + i ωα

βΛαβ , (149)

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19

with infinitesimal transformation parameters ǫα and ωαβ , as much as the H1 elements

h := ei vαβΛα

β ≈ I + i vαβΛα

β , (150)

with the also infinitesimal group parameter vαβ, and

b = e−i ξαPα , b ′ = e−i ( ξα+δξα )Pα , (151)

where the finite translational parameters ξα label the cosets b ∈ G/H1. The tildes in (151) are introduced to distinguish

the tilded b’s from the untilded b in (159) below. Using the Hausdorff–Campbell formula (B1) with the commutationrelations of the affine group

[Pα , Pβ ] = 0,

[Λαβ , Pγ ] = δαγ Pβ ,

[Λαβ ,Λ

γδ] = δαδ Λγ

β − δγβ Λαδ , (152)

we find the value of vαβ in (150) and the variation of the coset parameters ξα in (151) to be respectively

vαβ = ωα

β , δξα = −ξβωβα − ǫα , (153)

compare with the analogous PGT results (24). The nonlinear connection (11) is built in terms of the linear affineconnection

AM

:= −i(T)

ΓαPα − i(GL)

ΓαβΛα

β , (154)

whose components, the translational and the GL(4 , R) connection, transform respectively as

δ(T)

Γα = −(T)

Γβωβα +

(GL)

D ǫα , δ(GL)

Γαβ =

(GL)

D ωαβ . (155)

Replacing (154) in (11) we get

ΓM

:= b−1(d +A

M

)b = −i ϑαPα − i Γα

βΛαβ , (156)

where

ϑα :=(GL)

D ξα +(T)

Γα , Γαβ =

(GL)

Γαβ . (157)

As in the case of (151), we denote these objects with a tilde for later convenience. Applying (12), it is trivial to find

δϑα = −ϑβωβα , δΓα

β = Dωαβ , (158)

showing that the coframe ϑα in (157) transforms as a GL(4 , R) covector, in contrast to the linear translational

connection, see (155), while Γαβ remains unchanged as a GL(4 , R) connection.

So far, no metric tensor is derived from the gauging of the affine group. In order to deductively obtain a metricas a gauge–theoretical quantity, we have to consider a second nonlinear realization of G = A(4 , R) with auxiliarysubgroup H2 = SO(3 , 1). Being the homogeneous Lorentz group a (pseudo-)orthogonal group, it is equipped witha Cartan-Killing metric, namely the invariant Minkowski metric oαβ . When taken as the auxiliary subgroup of thenonlinear realization, the Lorentz group induces an automatic metrization of the theory. Certainly, as long as we onlyattend to the realization with H2 = SO(3 , 1), the metric can just be a constant, the Lorentz invariance δoαβ = 0 stillholding under gauge transformations of the whole affine group since, as a general feature of the nonlinear procedure,the total group G acts formally as its subgroup H2, see (8). Nevertheless, we are going to show how to establish thecorrespondence to the realization with H1 = GL(4 , R) studied above, in such a way that, by means of redefinitionsisomorphic to gauge transformations, ten Goldstone–like degrees of freedom may be either rearranged in the gaugepotentials or displayed as a variable metric tensor, depending on the nonlinear realization we consider, either withH2 = SO(3 , 1) or with H1 = GL(4 , R), see (167)–(170) below.

To study the case with H2 = SO(3 , 1), we start by splitting the generators of the general linear group into the sumof symmetric plus antisymmetric (Lorentz) parts as Λα

β = Sαβ + Lα

β. Then we apply the general formula (7) withthe particular parametrization

g = ei ǫαPαei αα

βSαβei βα

βLαβ , b := e−i ξαPαei hα

βSαβ , h := ei uα

βLαβ , (159)

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20

where the infinitesimal transformation parameters of the affine group are the translative ones ǫα and the generallinear parameters in (149), decomposed into symmetric plus antisymmetric contributions as ωα

β = ααβ + βα

β , whilethe infinitesimal nonlinear parameters uα

β correspond to the Lorentz subgroup H2. From (7) we find the variationsξ ′α = ξα + δξα and h′α

β = hαβ + δhα

β of the coset parameters of b in (159) to be respectively

δξα = −ξβ (αβα + ββ

α) − ǫα , δrαβ = (αα

γ + βαγ ) rγ

β − uγβ rα

γ , (160)

where we made use of the definition

rαβ :=

(eh)α

β := δαβ + hα

β +1

2!hα

γhγβ + · · · (161)

It is easy to check that, contrary to µαβ in (24), the nonlinear Lorentz parameters uα

β relevant for nonlinear trans-formations differ from the linear ones βα

β . But we do not need to know their explicit form, which can be calculatedfrom the vanishing of the antisymmetric part of the second equation in (160), see [26] [29].

The nonlinear connection corresponding to the choice H2 = SO(3 , 1) is obtained by replacing (154) into (11) takinginto account the decomposition Λα

β = Sαβ + Lα

β, with b as given in (159). We get

ΓM

:= b−1(d +A

M

)b = −i ϑαPα − iΓα

β(Sα

β + Lαβ

), (162)

with

ϑα :=((GL)

D ξβ +(T)

Γβ)rβ

α , Γαβ :=

(r−1

γ[ (GL)

Γγλ rλ

β − d rγβ]. (163)

The coframe ϑα in (163) transforms as a Lorentz covector, that is

δϑα = −ϑβ uβα , (164)

with uβα as the nonlinear Lorentz parameters, whereas the linear connection in (163), taken as a whole, behaves as

a Lorentz connection

δΓαβ = Duα

β . (165)

Observe however that, as read out from the r.h.s. of (162), the decomposition into two sectors of the Lie algebra ofGL(4 , R) gives rise to a splitting of the linear connection into the sum of a symmetric plus an antisymmetric part.Only the latter, with values on the Lorentz algebra, behaves as a true Lorentz connection, while the symmetric part(that is, the nonmetricity Qαβ := 2 Γ(αβ)) is a Lorentz tensor, varying as

δQαβ = 2 u(αγQβ)γ . (166)

Having completed the nonlinear realization of the affine group with the auxiliary subgroup H2, we are ready toestablish the correspondence between it and the one with H1. The affine objects of the approach with H1 = GL(4 , R)are displayed in (157), distinguished by tildes, while those of the H2 = SO(3 , 1) case are written without tildes in(163). By comparing (157) and (163) to each other, we find out that the relation between both kinds of quantities isisomorphic to a finite gauge transformation expressible as

ϑα =(GL)

D ξα +(T)

Γα = ϑβ(r−1

α , (167)

and

Γαβ =

(GL)

Γαβ = rα

γ[

Γγλ(r−1

)λβ − d

(r−1

)γβ], (168)

with the main difference that the matrix rαβ as given by (161) is not a gauge transformation matrix, but consists

of coset fields varying as shown in (160). It is precisely this peculiar transformation property of rαβ , involving both

the linear (ωαβ = αα

β + βαβ) as much as the nonlinear group parameters (uα

β), that is responsible for the differencebetween the gauge transformations –(158) versus (164), (165)– of the objects with and without tilde respectively,related to each other by rα

β as displayed in (167) and (168).

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21

In analogy to the latter equations, a correspondence can be established between the Minkowski metric oαβ , existingin the H2 = SO(3 , 1) approach as a natural invariant, and a correlated MAG-metric tensor gαβ defined in the contextof the approach with H1 = GL(4 , R) as

gαβ := rαµrβ

νoµν . (169)

The MAG-metric tensor (169) plays the role of a Goldstone field [21] [29]. Actually, the ten degrees of freedomassociated to it drop out by inverting the ”gauge transformation” (169), together with (167) and (168). In other words,it is possible to absorb the metric variables into redefined gauge potentials by using the nonlinear realization with theLorentz group H2 = SO(3 , 1) as the auxiliary subgroup instead of H1 = GL(4 , R). Accordingly, affine invariants canbe alternatively displayed in terms of explicit general linear quantities (with tildes), as in the standard formulationof MAG [16], or in terms of explicit Lorentz objects (without tildes) with the metric fixed to be Minkowskian. Forinstance, the line element can be doubly expressed as

ds2 = gαβϑα ⊗ ϑβ = oαβϑ

α ⊗ ϑβ . (170)

In accordance with the Goldstone–like nature of the MAG–metric, the field equations obtained by varying affineinvariant actions with respect to gαβ are known to be redundant [16]. However, we wont enter the study of detailsconcerning MAG–dynamics. The interested reader is referred to the literature, where quite general actions werestudied, involving quadratic curvature, torsion and nonmetricity terms, for which a number of exact solutions werefound [49] [50] [51] [52] [53] [54] [55] [56]. Discussions on the problem of the inclusion of matter sources in the MAGscheme can also be found for instance in references [57] [58] [59] [60] [61] dealing with phenomenological matter, andin [16] [27] [62] concerning fundamental matter.

VI. CONCLUSIONS

We presented a number of applications of NLR’s to the foundation of different gravitational gauge theories in orderto illustrate the variety of fields in which the method reveals to be useful, providing underlying mathematical unityand simplicity. In particular, it is worth to recall once more that the Hamiltonian approach developed in Section IV

in terms of PGT connection variables revealed to be dynamically equivalent to a theory of the Ashtekar type. (Sothat, conversely, the latter can be regarded as a reformulation of the Hamiltonian Poincare gauge theory built fromthe Einstein–Cartan action.)

As a general result derived from the different examples studied by us, we want to remark that thanks the NLR’sthe description of interactions is achieved exclusively in terms of connections, in accordance with the general gauge–theoretical program. Neither the coframes nor the MAG–metric are to be regarded as separate gravitational potentialsof specific nature, but rather as ordinary Yang-Mills objects. Indeed, the coframes are interpreted as a kind of gaugepotentials, namely as nonlinear translative connections, while the metric of MAG is found to be a Goldstone fieldplaying no fundamental physical role, since its degrees of freedom can be transferred to redefined gauge potentials.In the limit of vanishing Poincare connections (corresponding to zero gravitational forces), the tetrads (27) reduce tothe special relativistic ones ϑα = dξα, the fields ξα playing the role of ordinary coordinates –as read out from theirtransformations (24)–, so that the Minkowski space of Special relativity can be seen as the residual structure left bythe dynamical theory of spacetime when gravitational interactions are switched off.

Matter sources in the context of NLR’s were exemplified by Dirac fields in PGT, whose coupling to translationsgives rise to a background fermion mass contribution. Instead, the inclusion of fundamental matter in the context ofnonlinear metric–affine gravity remains only partially explored. A further natural extension of the nonlinear methodnot yet developed consists in its application to mechanisms of spontaneous symmetry breaking –from G to a residualsymmetry H ⊂ G– in the case of external as much as of internal groups. Let us also hope that, although restrictedfor the time being to classical aspects of gravity, the nonlinear framework can become an useful tool to deal withquantum aspects of gravitational gauge theories.

Acknowledgments

The authors want to thank F.W. Hehl and E.W. Mielke for permanent interest and encouragement along the years.

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22

APPENDIX A: ETA BASIS

The objects constituting the eta basis defined in the present Appendix, built as the Hodge duals of exterior productsof tetrads [16], are convenient to simplify the notation when dealing with differential forms. In terms of (27) we definethe Levi-Civita object (that is, the 0-form element of the eta basis) as

ηαβγδ := ∗(ϑα ∧ ϑβ ∧ ϑγ ∧ ϑδ ) , (A1)

and with the help of it the 1-form element

ηαβγ := ∗(ϑα ∧ ϑβ ∧ ϑγ ) = ηαβγδ ϑδ , (A2)

the eta-basis 2-form element

ηαβ := ∗(ϑα ∧ ϑβ ) =1

2!ηαβγδ ϑ

γ ∧ ϑδ , (A3)

the 3-form element (dual of the tetrad)

ηα := ∗ϑα =1

3!ηαβγδ ϑ

β ∧ ϑγ ∧ ϑδ , (A4)

and the 4-form of the eta-basis, or four–dimensional volume element

η := ∗1 =1

4!ηαβγδ ϑ

α ∧ ϑβ ∧ ϑγ ∧ ϑδ . (A5)

The exterior product of the coframe ϑµ with the elements (A1)–(A4) of the eta basis yields respectively

ϑµ ∧ ηαβγδ = −δµα ηβγδ + δµδ ηαβγ − δµγ ηδαβ + δµβ ηγδα , (A6)

ϑµ ∧ ηαβγ = δµα ηβγ + δµγ ηαβ + δµβ ηγα , (A7)

ϑµ ∧ ηαβ = −δµα ηβ + δµβ ηα , (A8)

ϑµ ∧ ηα = δµα η . (A9)

For an arbitrary p–form α on four–dimensional space with Lorentzian signature, the double application of the Hodgedual operator reproduces α itself up to the sign as ∗∗α = (−1)p(4−p)+1 α. A further relation involving Hodge dualityreads ∗(α ∧ ϑµ ) = eµ⌋ ∗α, while for differential forms α, β of the same degree p, equation ∗α ∧ β = ∗β ∧ α holds.The eta basis (A1)–(A5) and the algebraic relations of the present Appendix are extensively used thorough the wholework.

APPENDIX B: HAUSDORFF–CAMPBELL FORMULAS

In order to make the present exposition as self contained as possible, we give the well known formulas

e−ABeA = B − [A ,B ] +1

2![A , [A ,B ] ] − 1

3![A , [A , [A ,B ] ] ] + ... (B1)

e−AdeA = dA− 1

2![A , dA ] +

1

3![A , [A , dA ] ] − ... (B2)

eA+δA = eA + δeA +O(

(δA )2)

= eA(1 + e−AδeA

), (B3)

useful for checking calculations.

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APPENDIX C: NONLINEAR REALIZATION OF THE POINCARE GROUP WITH SO(3) ASAUXILIARY SUBGROUP

By decomposing the Lorentz generators Lαβ into boosts Ka and space rotations Sa , defined respectively as

Ka := 2La0 , Sa := −ǫabcLbc (a = 1 , 2 , 3) , (C1)

the commutation relations (22) transform into

[Sa , Sb ] = −i ǫabcSc ,

[Ka ,Kb ] = i ǫabcSc ,

[Sa ,Kb ] = −i ǫabcKc ,

[Sa , P0 ] = 0 ,

[Sa , Pb ] = −i ǫabcPc ,

[Ka , P0 ] = i Pa ,

[Ka , Pb ] = i δabP0 ,

[Pa , Pb ] = [Pa , P0 ] = [P0 , P0 ] = 0 . (C2)

In the nonlinear transformation law (7), we take the infinitesimal Poincare group elements g ∈ G, and the SO(3)group elements h ∈ H , to be respectively

g = ei ǫαPαei β

αβLαβ ≈ 1 + i(ǫ0P0 + ǫaPa + ζaKa + θaSa

)(C3)

and

h = eiΘaSa ≈ 1 + iΘaSa , (C4)

and further we parametrize b ∈ G/H as

b = e−i ξαPαei λaKa , (C5)

being ξα and λa finite coset fields. Eq.(7) with the particular choices (C3)–(C5) yields on the one hand the variationof the translational parameters

δξ0 = −ζaξa − ǫ0 , (C6)

δξa = ǫabcθbξc − ζaξ0 − ǫa , (C7)

which, since ζa := βa0 and θa := − 12ǫ

abc β

bc as read out from (C3), can be rewritten as

δξα = −ββα ξβ − ǫα , (C8)

compare with (24), showing that the coset parameters ξα associated with the translations behave in fact as coordinates.On the other hand, the variations of the boost parameters in (C5) turn out to be

δλa = ǫabcθbλc + ζa|λ| coth |λ| +

λaλbζb

|λ|2 (1 − |λ| coth |λ| ) , |λ| :=√λaλa . (C9)

Instead of dealing with λa, it is preferable to introduce the velocity fields

βa := − λa

|λ| tanh |λ| , γ :=1√

1 − β2, (C10)

varying as

δ γ = − ζa (γ βa ) , (C11)

δ (γ βa ) = ǫabc θb (γ βc ) − ζa γ , (C12)

that is, as the components of a Lorentz four–vector (γ , γ βa ) , as can be easily checked by comparing (C11), (C12)with (C6), (C7).

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24

Finally, (7) also enforces Θa in (C4) to be

Θa = θa +γ

(1 + γ )ǫabc ζ

b βc . (C13)

According to (8), Θa is the modified SO(3) gauge parameter throw which the nonlinear action of the Poincare grouptakes place as

δψ = iΘaρ (Sa)ψ (C14)

on fields ψ of representation spaces of SO(3), being ρ (Sa) the corresponding representation of the SO(3) generators.Now we turn our attention to the nonlinear gauge fields (11), defined from the ordinary linear Poincare connection

AM

:= −i(T )

ΓαPα − iLor

ΓαβLαβ , (C15)

standing(T )

Γα for the translational andLor

Γαβ for the Lorentz contribution. (Although modified by this additionalspecification, (C15) is identical with (25).) Making use of (C1) we introduce for (11) the notation

ΓM

= −i ϑαPα − i ΓαβLαβ = −i ϑ0P0 − i ϑaPa + iXaKa + i AaSa , (C16)

where a simple application of the Hausdorff-Campbell formulas of Appendix B yields

ϑα = ϑβbβα , (C17)

Xa := Γ0a = (b−1)0µ(

Lor

Γµνbν

a − d bµa

), (C18)

Aa :=1

2ǫabcΓ

bc =1

2ǫabc (b−1)bµ

(Lor

Γµνbν

c − d bµc

), (C19)

expressed with the help of the boost matrix

b00 = (b−1)0

0 := γ , b0a = −(b−1)0

a := −γβa ,

ba0 = −(b−1)a

0 := −γβa , bba = (b−1)b

a := δab + (γ − 1)βbβ

a

β2, (C20)

built from the fields (C10). The Lorentz covectors

ϑα :=Lor

D ξα +(T )

Γα , (C21)

in the r.h.s. of (C17) are identical with the Lorentz coframes (27) of the nonlinear approach studied above. (Theabbreviation Lor over the covariant differentials in (C21) indicates that they are constructed with the linear Lorentz

connectionLor

Γαβ in (C15).) Despite the formal analogy of (C17)–(C19) with gauge transformations, in fact the cosetparameters λa, and thus (C10) and (C20), are fields of the theory rather than gauge parameters. Consequently,(C17)–(C19) are definitions of new variables whose transformation properties depend on (C11), (C12). Actually,

while ϑα in (C21) transforms as a Lorentz covector andLor

Γαβ in (C15) as a Lorentz connection, for the quantitiesdefined in (C17)–(C19) we find

δϑ0 = 0 , (C22)

δϑa = ǫabc Θb ϑc , (C23)

δXa = ǫabc ΘbXc , (C24)

δAa = −DΘa := −(dΘa + ǫabcA

b Θc). (C25)

That is, the tetrads become split into an SO(3) singlet –the invariant time component ϑ0 – plus an SO(3) covector

–the triad ϑa–. The nonlinear boost connection 1–forms Xa also transform as the components of an SO(3) covector.Only the SO(3) connections Aa retain their connection character. The nonlinear field strength (16) built from (C16)reads

F := dΓM

+ ΓM

∧ ΓM

= −i T 0P0 − i T aPa + i (DXa)Ka + iRaSa , (C26)

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25

where we introduce the definition of the torsion

T 0 := d ϑ0 + Γµ0 ∧ ϑµ = d ϑ0 + ϑa ∧Xa , (C27)

T a := d ϑa + Γµa ∧ ϑµ = D ϑa + ϑ0 ∧Xa , with D ϑa := d ϑa + ǫabcA

b ∧ ϑc , (C28)

the boost curvature

DXa := dXa + ǫabcAb ∧Xc , (C29)

and the rotational curvature

Ra := F a − 1

2ǫabcX

b ∧Xc , with F a := dAa +1

2ǫabcA

b ∧ Ac , (C30)

respectively. It is trivial to check

DXa = R0a , Ra =1

2ǫabc R

bc , (C31)

relating (C29), (C30) to the four–dimensional curvature Rαβ := dΓα

β + Γγβ ∧ Γα

γ , with the same form as (31) but

built from the Lorentz connection Γαβ in (C16).

APPENDIX D: FOLIATION OF SEVERAL POINCARE OBJECTS

In the present Appendix we apply the foliation procedure of Subsection IV. B to the quantities introduced inAppendix C. Regarding the fundamental objects (C17)–(C19), notice that trivially the zero component of the tetrad

(C17), with the form ϑ0 = u0 d τ as in (51), only includes a longitudinal contribution, whereas ϑa = ϑa

only contains atransversal one. On the other hand, the boost nonlinear connection (C18) and the SO(3) connection (C19) decompose

as Xa = ϑ0Xa⊥ +Xa and Aa = ϑ0Aa

⊥ +Aa respectively. Furthermore, the decomposition of the torsion components(C27), (C28) takes the form

T 0 = −ϑ0 ∧(d log u0 +Xa

⊥ϑa

)+ ϑa ∧Xa , (D1)

T a = ϑ0 ∧(

Le0ϑa +Xa

)+D ϑa , (D2)

where the covariantized Lie derivative and the transversal part of the covariant differential are respectively defined as

Le0ϑa := e

0⌋Dϑa = le

0ϑa + ǫabcA

b⊥ϑ

c , D ϑa := d ϑa + ǫabcAb ∧ ϑc. (D3)

The boost curvature (C29) splits into longitudinal and transversal parts as

DXa = ϑ0 ∧[

Le0Xa − 1

u0D

(u0Xa

)]+DXa , (D4)

in terms of the covariant derivatives

Le0Xa := e

0⌋DXa = le

0Xa + ǫabcA

b⊥X

c , (D5)

D(u0Xa

):= d

(u0Xa

)+ ǫabcA

b(u0Xc

), (D6)

DXa := dXa + ǫabcAb ∧Xc , (D7)

compare with (56). The SO(3) curvature (C30) and its Hodge dual, see (55), decompose respectively as

Ra = ϑ0 ∧Ra⊥ + Ra , ∗Ra = ϑ0 ∧ #Ra − #Ra

⊥ , (D8)

with definitions

Ra⊥ := F a

⊥ − ǫabcXb⊥X

c , F a⊥ := le

0Aa − 1

u0D

(u0Aa

), (D9)

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26

and

Ra := F a − 1

2ǫabcX

b ∧Xc , F a := dAa +1

2ǫabcA

b ∧ Ac . (D10)

In order to complete the set of foliated objects needed in Section IV, we give here the 3+1 decomposition of the4-dimensional eta basis of Appendix A as

ηabc = −ϑ0 ηabc , ηab = ϑ0 ∧ ηab , ηa = −ϑ0 ∧ ηa , η = ϑ0 ∧ η , (D11)

where the bar over the etas in (D11) means their restriction to the three–space as

ηabc := η0abc = ǫabc ,

ηab := η0ab = ǫabcϑc ,

ηa := η0a =1

2ǫabcϑ

b ∧ ϑc ,

η := η0 =1

3!ǫabcϑ

a ∧ ϑb ∧ ϑc . (D12)

The identification of ηabc with the group constants ǫabc of SO(3) in (D12) is possible due to the fact that, being the

holonomic SO(3) metric the Kronecker delta, one has ηabc = #(ϑa ∧ ϑb ∧ ϑc

)=

√det(δmn) ǫabc = ǫabc .

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